In this unit, we will primarily be interested in understanding determinants. Matrices will be explored in a separate unit. However, since there are some similarities and overlap in concepts, this page provides a quick overview of what both matrices and determinants are, and how they are different.

Matrices

In this section you will see the term "matrix" and the term "matrices." Matrices is the plural of matrix.

A matrix is nothing more than an array (rows and columns) of numbers, written with square brackets (or large parentheses) around them. The following is an example of a matrix:

Here is another example:

Note that the number of rows and columns do not have to match. In fact, we could have a matrix with just one row, or a matrix with just one column:

We will most commonly use capital letter variables to represent matrices. It is important to remember that a matrix is just a collection of numbers in columns and rows; the matrix itself does not have a value; it is a collection of values. Any time you create a table of values in rows and columns, it could be written in a consolidated way using a matrix.

The table above is really just a matrix:

Every matrix has dimensions, and the dimensions are a way of describing how big the matrix is. If we say that a matrix is a 2 x 3 matrix, that means it has 2 rows, and 3 columns.  The matrix above is a 4 x 2 matrix, because it has 4 rows and two columns.

We could refer to any entry of the matrix using a variable and a subscript. For example, a_1,2 is the element in the first row, second column, and a_3,1 is the element in the third row, first column. In the matrix above, a_1,2 = 85, and a_3,1 = 11.

Determinants

A determinant looks a lot like a matrix, but it is, actually, quite different. Here's an example of a determinant:

If you look at this, and compare it to matrix A, you will observe that the only difference between them is that the matrix has square brackets, and the determinant has straight line bars around it. But a matrix and a determinant are very different, even though they look very similar. Unlike a matrix, a determinant isn't just an array of numbers; it also has a value, which can be calculated using rules you'll be taught in the next session. The other important difference to take note of now is that even though in a matrix, the number of rows doesn't have to match the number of columns, in a determinant, they must match. Another way of saying this: all determinants are square. You can have a 2 x 2 determinant, a 3 x 3 determinant, a 4 x 4 determinant (and so forth) but you cannot have a 2 x 3 determinant.

Every square matrix has an associated determinant made up of all of its entries. In fact, we often refer to determinants as "the determinant of matrix X" or "the determinant of matrix Y."

In the next section we'll explore how to evaluate (find the value of) a 2 x 2 determinant.